Overflow analysis of a fixed-point implementation of the Goertzel algorithm

It is shown that the second-order Goertzel algorithm, though favored over the first-order Goertzel algorithm for its reduced computational complexity, is in fact prone to overflows when implemented in hardware with only fixed-point arithmetic. An overflow analysis reveals that the second-order Goertzel algorithm requires a different scaling factor at each frequency sample to systematically eliminate the possibilities of overflow